I've dealt with some of these instances in the past, like SS getting punished for me saying hi to him on the street (we live in a VERY small town). Also, PBFH going off on DH when SS wanted us to come watch his Rainguttter Regatta, telling him that if we went, she would not allow SS to take his boat.
Question: how involved with her schooling are you? If not very, change that. If you have joint legal, there is NOTHING the BM can do to stop you from participating in ANY way. Talk to her teachers and ask them if they've seen any 'off behaviors' or if she's mentioned being threatened with punishment or been beaten in the past. They are all amndatory reporters and are required by law to report ALL suspected abuse.
As for the cops and the DUI, I'd be contacting Family Services myself. Cops are also mandatory reporters. I am in EMS and am one also. In my state, if it is found that I knew of suspected abuse but did not report it, the penalties are SEVERE, steep fines and jail time both. I would not be surprised if your state has something similar. These cops MUST be reported as she put her child in IMMINENT danger and they did nothing about it.
Bottom line: you need to teach your child whom she can go to to talk to that's an objective 3rd party (teacher, school counselor, minister, etc.). Her telling you isn't enough, because if only you are reporting it, it will look like a custody dispute. Objective 3rd party reports are much more readily accepted.
If you have any visitation at all, see if you can arrange for counseling during the time you have her. I have my son currently with a therapist who works both nights and weekends (we've seen her on a Sat. and his appts. are at 6 pm.), so it can be done, you just have to dig deeper.
As for the sticker, notify the base and tell them it is no longer your car and you are not responsible for it, then give them her name and address. If she bitches to you, tell her they probably got her info from her plate number. But if she does something screwy on base, it will fall back on you, and they don't take too kindly to that crap anymore.